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A Multi-Frequency Input-Admittance Model of Locomotive Rectifier Considering PWM Sideband Harmonic Coupling in Electrical Railways

Xiangyu Meng, Zhigang Liu, Guorong Li, Xunjun Chen, Siqi Wu, Keting Hu

arXiv:2607.09275v1eess.SY

TL;DR

High-frequency railway harmonic instability is not fully captured by traditional small-signal averaging models because they omit PWM sideband harmonics. This paper develops a multifrequency input-admittance model and converts it to a SISO model, finding that sideband harmonics dominate above half the switching frequency and validating stability influences with HIL results.

  • Problem

    Traditional converter small-signal averaging models are effective below 1/2 switching frequency because they ignore PWM sideband harmonic components, while analysis of harmonic instability remains insufficient.

  • Method

    The paper analyzes perturbation and sideband-frequency propagation through PWM, derives a multifrequency locomotive rectifier input-admittance model, and converts it to a SISO model while preserving sideband coupling.

  • Results

    PWM sideband harmonic components dominate locomotive rectifier input-admittance characteristics above 1/2 switching frequency, and HIL results match the model's stability analyses across switching-frequency conditions.

  • Takeaways & Limitations

    The derived SISO model supports high-frequency stability analysis and reveals the effects of switching frequency, control bandwidth, and traction-network impedance on system stability.

Abstract

from arXiv · show

Electrical railway harmonic instability issues are common in the high-frequency range. The effective frequency of the traditional converter's small-signal averaging model is below 1/2 switching frequency since the pulse width modulation (PWM) sideband harmonic components are ignored. In this article, the dynamic propagations of perturbation frequency and the generated PWM sideband components are constructed first. Then the locomotive rectifier's multi-frequency input-admittance model is derived appropriately. Afterward, an admittance conversion approach is used to convert the multi-frequency model into the single-input-single-output (SISO) model whereas retaining the sideband frequency couplings. The proposed SISO model is more accurate than the traditional small-signal averaging model in the frequency range higher than 1 / 2 switching frequency. It is found that PWM sideband harmonics dominate the locomotive rectifier's input-admittance characteristic higher than 1 / 2 switching frequency. Finally, based on the proposed model, the influence of different switching frequencies, control bandwidths, and traction network impedance on system harmonic stability is revealed by the hardware-in-the-loop (HIL) results.

I. INTRODUCTION

High-frequency harmonic-instability analysis in locomotive–traction-network systems is limited because traditional small-signal averaging models ignore PWM sideband harmonics above half the switching frequency. The paper develops coupled multi-frequency and SISO admittance models to improve this analysis and examines key stability influences.

  • Locomotive rectifiers interact nonlinearly with traction networks, producing high-order harmonic resonance and harmonic instability that can amplify network voltage and current.
  • Traditional small-signal averaging models ignore PWM-generated sideband harmonics, limiting their effective frequency range to below 1/2 switching frequency.
  • A multi-frequency locomotive input-admittance model and converted SISO model retain PWM sideband coupling and improve accuracy above 1/2 switching frequency.
  • The paper derives a three-order PWM transfer-function matrix to characterize propagation of perturbation and sideband harmonic frequencies.
  • The derived model analyzes how switching frequency, control bandwidth, and traction impedance affect L-N system harmonic stability, with HIL testing included.

II. SYSTEM DESCRIPTION

The study models an all-parallel autotransformer traction network comprising substations, conductors, transformers, and locomotives. Its equivalent impedance and distributed capacitance are represented for L-N system analysis.

  • The all-parallel AT traction network includes a traction substation, traction network, AT transformers, locomotives, contact wire, rail wire, and AT feeder.
  • The generalized symmetrical-components model represents the traction network while converting parameters to the onboard transformer’s low-voltage side.
  • The equivalent network impedance incorporates traction-transformer impedance, AT leakage reactance, sequence impedances, conductor distances, and onboard-transformer turns ratio.
  • Distributed capacitance is included through contact-wire, AT-feeder, rail-wire, protection-wire, and ground-wire capacitances to ground.
  • The resulting traction-network impedance is calculated from passive network components using the stated equivalent-circuit relation.

III. MODEL OF LOCOMOTIVE INPUT-ADMITTANCE

The locomotive rectifier control system contains PLL, voltage, current, and PWM blocks, but high-frequency admittance modeling emphasizes PWM effects because low-bandwidth PLL and voltage-control dynamics are less influential there.

  • The rectifier control structure includes a phase-locked loop, direct voltage control, alternating current control, and PWM block.
  • PLL and direct-voltage-control blocks mainly affect low-frequency input impedance because their control bandwidths are relatively low.
  • Modulation delay and PWM-generated sideband harmonics significantly influence locomotive input admittance in the high-frequency range.
  • For the harmonic-instability analysis, PLL and direct-voltage-control blocks can be omitted because they have limited influence at high frequencies.

A. MODEL OF DIGITAL PWM COMPARATOR

The digital PWM model tracks how a perturbation frequency propagates into the output spectrum, including sideband harmonics and their frequency coupling. A three-order transfer-function matrix is validated against measurements and supports modeling beyond half the switching frequency.

  • The digital PWM comparator uses bipolar asymmetric regular sampling with sampling at the triangle carrier peak, giving fsa = 2fsw.The model includes sampling and computation delays in the digital process.
  • The PWM output contains the perturbation, fundamental, switching-frequency harmonics, and sideband components generated by PWM nonlinearity.For a 420 Hz perturbation, the output spectrum includes the perturbation frequency, 50 Hz fundamental, inherent switching harmonics, and sidebands.
  • The analyzed frequency range is limited to the Nyquist frequency, so only fpwmb1 and fpwmb2 sidebands are retained.The higher-order sidebands fpwmb3 and fpwmb4 are ignored to avoid aliasing beyond the Nyquist frequency.
  • G0 maps the PWM reference perturbation to the same output frequency, while G1 and G2 map it to sideband components.These mappings describe the dynamic frequency coupling in the PWM process.
  • The PWM process is represented by a three-order transfer-function matrix that propagates the perturbation and its sideband frequencies.The matrix includes reference and output components at fp, fpwmb1, and fpwmb2, together with the corresponding G0, G1, and G2 mappings.
  • Measured responses agree well with the theoretical G0, G1, and G2 models up to 2fsw.Modeled results are shown as solid lines and measured results as circles.

B. MULTI-FREQUENCY INPUT-ADMITTANCE MODEL OF LOCOMOTIVE RECTIFIER

The locomotive rectifier is modeled with a multi-frequency input admittance that includes perturbation and PWM sideband components. Around 560 Hz, sideband harmonics become dominant above the separatrix, where the traditional averaging model omits their influence.

  • The rectifier control model represents onboard transformer admittance and ACC PI regulation as diagonal linear-element matrices.
  • The derived input-admittance model is a multi-frequency matrix covering the perturbation frequency and two PWM sideband harmonics.
  • The first term is the small-signal averaging model, while the second and third terms represent PWM sideband effects as additional parallel admittance.
  • 560 Hz marks the separatrix, approximately equal to half the switching frequency, separating perturbation-dominated and sideband-dominated admittance behavior.
  • Above the separatrix, PWM sideband admittance magnitudes exceed the perturbation-harmonic magnitude, and their gap increases farther from the separatrix.
  • The traditional averaging model is ineffective above half the switching frequency because it overlooks PWM sideband harmonic components.

C. EQUIVALENT SISO INPUT-ADMITTANCE MODEL OF LOCOMOTIVE RECTIFIER

The multi-frequency rectifier model is converted into an equivalent SISO model so frequency-domain stability analysis remains intuitive while retaining PWM sideband coupling. Compared with the averaging model, it better matches measurements in the mid- and high-frequency ranges.

  • The MIMO impedance model is converted into a SISO model to simplify frequency-domain stability analysis while retaining PWM sideband harmonic coupling.
  • The conversion uses parameters α, c, and b from the rectifier admittance matrix and Q from the traction-network impedance matrix.
  • 700–1000 Hz: the SISO model fits measured admittance results better than the small-signal averaging model.
  • 0–100 Hz: SISO-model accuracy is low because the PLL and voltage outer loop are omitted from the model.

D. INPUT-ADMITTANCE MODEL OF LOCOMOTIVE

The locomotive model aggregates four identical, independently operated power units connected in parallel. Its electrical chain steps the traction-network voltage down, rectifies it to a constant dc voltage, and represents the inverter and motor as a constant load.

  • One locomotive contains four identical power units connected in parallel and operated independently.
  • The onboard transformer steps 27.5 kV down to 950 V, which the rectifier converts into 1800 V dc.
  • The dc voltage remains constant at 1800 V because of the dc-side filter capacitance.
  • Under constant-power operation, the inverter and traction motor are represented by an equivalent constant load resistance.
  • The locomotive input admittance is expressed as Y_l = 4Y_siso for its four parallel power units.

IV. HIL SIMULATION AND VERIFICATION

The HIL platform verifies the locomotive and traction-network model using four identical power units, real-time simulation, and measured PCC signals. ADC sampling and computed PWM pulses connect the controller to the real-time simulator.

  • The HIL circuit model represents the locomotive with four identical power units and the traction network as a separate modeled subsystem.
  • The platform uses a NI-PXIe-FPGA-7868R real-time simulator, host PC, oscilloscope, and StarSim software.
  • PCC voltage and current are sampled by the real-time controller ADC at f_sa = 2f_sw.
  • Calculated PWM pulses are sent to the real-time simulator to control the locomotive rectifier, while PCC waveforms are displayed and analyzed on an oscilloscope.

A. INFLUENCE OF SWITCHING FREQUENCY ON SYSTEM STABILITY

Lowering the switching frequency reduces the locomotive–network system’s stability margin and can produce harmonic instability. The HIL waveforms corroborate the model’s predicted transition from stable operation to instability.

  • At fsw = 1000 Hz, the impedance-ratio Nyquist diagram remains far from the critical point, indicating a wide stability margin.
  • At fsw = 800 and 600 Hz, the Nyquist diagram moves progressively closer to the critical point, while fsw = 500 Hz produces instability.
  • When fsw decreases from 1000 to 600 Hz, HIL voltage and current waveforms become distorted, indicating critical stability.
  • When fsw decreases from 1000 to 500 Hz, voltage and current harmonics are considerably amplified and the system becomes unstable because of unstable PCC voltage.
  • Lower switching frequencies increase sampling delay and can create a more negative high-frequency damping range in the locomotive input admittance.
  • The control bandwidth should be below 10% of the switching frequency to weaken PWM sideband harmonics and improve control performance.

B. INFLUENCE OF CONTROL BANDWIDTH AND TRACTION NETWORK IMPEDANCE ON SYSTEM STABILITY

Increasing control bandwidth and varying traction-network impedance change the system’s stability margins. The model and HIL results identify bandwidth-related high-frequency instability and confirm the model’s analysis.

  • The intersection points of locomotive and traction-network admittances evaluate stability using phase difference below 180° and phase margin above 0°.
  • Increasing control bandwidth from 1 to 5 p.u. reduces the first intersection point’s phase margin from 111.8° to 63.7°.
  • At 8 p.u. bandwidth, Ysiso(s) is below −90° above 630 Hz and the phase margin becomes negative, causing high-frequency HIS in that range.
  • Under changing control bandwidth, HIL waveforms show generated harmonics mainly around 650 Hz, with 300 Hz components identified as PWM sideband harmonics.
  • The HIL results confirm the developed impedance model and its stability-analysis results under varying control bandwidths.
  • With traction-network capacitance at 200 μF and 100 μF, the voltage and current remain stable without newly generated harmonics in the reported intervals.
  • Parameter variations produce different stability margins, and the established input-admittance model can analyze the resulting harmonic-instability issues.

V. CONCLUSION

The paper models PWM sideband coupling in locomotive input admittance and converts the resulting multifrequency representation into a high-frequency SISO model. The model identifies sideband-dominated behavior and reveals how system parameters affect harmonic stability.

  • PWM comparator dynamics produce MIMO locomotive input-admittance characteristics through propagation of perturbation and sideband harmonic frequencies.
  • A multifrequency input-admittance model and converted SISO model retain PWM sideband coupling for high-frequency analysis.
  • PWM sideband harmonics dominate locomotive input-admittance characteristics when perturbation frequency exceeds half the switching frequency.
  • The model reveals the effects of switching frequency, control bandwidth, and traction-network impedance on system stability and identifies unstable harmonic components.
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